For a generic metric on an $m$-dimensional the manifold the eigenvalues of the Laplacian are all simple.  Fix such a metric and denote the coresponding eigenvalues by


$$ \lambda_1, \lambda_2,\cdots $$


Using Weyl's asymptotic expansion we conclude that 

$$\lambda_n\sim \text{const}\cdot n^{m/2}. $$

Thus for any polynomial $P$ of degree $d>m/2$ we have


$$ \lim_{n\to\infty} \lambda_n/P(n) = 0. $$